# AUBURN MECH 7220 - Laminar External Flow (7 pages)

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## Laminar External Flow

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- School:
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- Course:
- Mech 7220 - Convection Heat Transfer

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5 Laminar External Flow 5 1 The boundary layer A boundary layer is an easy physical concept to grasp it is the result of the no slip boundary condition as fluid flows over the surface of an object Because of viscosity the fluid directly adjacent to the object is sheared and this effect is confined to a relatively thin layer parallel to the surface i e the boundary layer A boundary layer is also a mathematical concept and boundary layer behavior can occur whenever the higher order derivatives in a differential equation are multiplied by relatively small parameters This statement is more arcane yet it is important in terms of gaining insight into why physical boundary layers are thin and what solution methods can be used to solve for the transport of momentum and heat across the boundary layers A simple example both physical and mathematical of boundary layer behavior can be seen in a 1 D convective diffusive problem Say a uniform 1 D flow enters a control volume with temperature T0 At some distance L downstream the flow is brought to a new temperature T1 All the while the density and the velocity u stay constant The boundary at x L might represent for example a porous surface which is maintained at T1 The steady 1 D energy equation for this situation is U C Denote non dimensional variables as T dT d2 T k 2 dx dx T T1 T0 T1 1 x x L 2 and the DE becomes dT 1 d2 T dx P eL dx2 T 0 1 T 1 0 with P eL 3 4 uL C uL k The solution to this problem is T x P eL 5 eP eL eP eL x eP eL 1 6 A plot of the results is shown below T T1 T0 T1 1 PeL 1 0 8 PeL 100 PeL 10 0 6 0 4 PeL 0 1 0 2 0 0 0 2 0 4 0 6 0 8 1 x L Note that as P eL increases the temperature distribution goes from being linear to increasingly piled up into a narrow region adjacent to the x L boundary Indeed for P eL 100 the temperature is basically 1 uniform at T 1 except for a small region of steep gradient at the outflow surface This region of steep gradient would be considered a boundary layer The 2nd order derivative for this case is multipled by the small parameter 1 100 0 01 This might lead one to assume that the 2nd derivative term could be neglected from the DE and it could through most of the flow region Getting rid of this term would result in dT 0 T constant 1 7 dx where the constant is evaluated from the first BC However the problem has two boundary conditions and the zero temperature condition must be maintained at the outflow wall A first order DE cannot accommodate two boundary conditions so somewhere the second order derivative term must become significant It will become significant in the neighborhood of the T 0 surface The thickness of the boundary layer for this simple problem can be determined by reformulating the problem Reverse the coordinate direction so that x runs from the T 0 surface and define a scaled or stretched coordinate x via x x P eL 8 The scaled DE becomes dT d2 T dx dx 2 T 0 0 T x 1 9 10 In this view the far surface with T 1 now exists at x the model describes only the behavior within the boundary layer The solution is now T x 1 e x 11 and this indicates that the temperature reaches T 0 99 for x 4 61 The thickness of the boundary layer is therefore 4 61 L 4 61 bl 12 P eL u Some points to make regarding the boundary layer effect are 1 The boundary layer represents the region in which all derivatives in the governing DE contribute significantly to the transport process Outside of this region transport is dominated by a single mechanism convection in this case 2 The details of the boundary layer i e the thickness can be neglected if one is interested in predicting only the bulk characteristics of the flow For the simple model examined here the bulk or freestream characteristics would be predicted by Eq 7 and this result gives the conditions at the edge of the boundary layer 3 On the other hand the details of the boundary layer are absolutely critical if one wishes to predict the rate of transport to from a surface that is exposed to a flow Note that Eq 7 cannot predict the rate of heat transfer to the T 0 surface 4 On an order of magnitude level the rate of transport across the boundary layer will be q 00 k T Ts bl 13 This approximation does not have much relevance to the simple 1 D model because the heat flux must be q 00 u C T1 T0 by virtue of the first law for the system and this is precisely what the above equation would give It will have more bearing in 2 D flow situations 2 5 2 Momentum transfer on a flat plate Consider a situation in which a flat plate is exposed to a flow that runs parallel to its surface Far from the surface of the plate the velocity is u A boundary layer will form at the leading edge of the plate and the thickness of the layer will grow with distance downstream Everywhere the flow remains laminar the conditions for laminar BL flow will be discussed below The objective is to predict the velocity profile in the boundary layer Assuming incompressible flow the continuity and momentum equations for the situation are u v 0 x y u 1 dP 2u u v 2 u x y dx y 14 15 where x y are the streamwise and normal coordinates and u v are the associated components of velocity Assumptions are 1 there is negligible pressure gradient in the normal direction so that P P x 2 streamwise diffusion of momentum is negligible relative to streamwise convection Outside the boundary layer the flow is 1 D in the x direction so u u constant and v 0 The momentum equation applied in this region would show that dP dx 0 The pressure gradient term will be neglected from here on out yet it will become relevant i e non zero whenever the external flow is accelerating or decelerating as would be the case for a surface that is tilted with respect to the flow 5 2 1 Order of magnitude analysis It is possible to use Eqs 14 and 15 to estimate how the boundary layer thickness changes with position x from the leading edge At some point x at which the boundary layer is thick the change in u is at most u over the distance x The normal component of velocity goes from 0 to some characteristic value v over the normal distance y An order of magnitude analysis of the continuity …

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